Randomness and Probability

[Demagog]

I've stayed up far too long today, so I just had one of my crazy ideas. I'd like to see how long it takes for a random number generator, on average, to guess a correct number when it is used to play the "higher or lower" game, at different ranges of starting numbers. An example:

I choose the number 63, out of the numbers 1 through 100.
I set the RNG to pick a random number from 1 to 100.
It outputs 41.
I set it to pick a number from 42 to 100.
It outputs 88.
I set it to pick a number from 42 to 87.
It outputs 55.
I set it to pick a number between 56 and 87.
It outputs 85.
I set it to pick a number between 56 and 84.
It outputs 63.

It took the RNG five guesses to output the correct number.

I'd like to do this experiment a large number of times with several different ranges of numbers, in order to see if there is a correlation between the size of the range and the number of guesses the RNG takes. The hypothesis is that there is.

So, a list of ranges to test:

1-10
1-25
1-50
1-100
1-250
1-500
1-1000
1-2500
1-5000
1-10000

Each would be tested a large number of times (any idea how many?) and then the guesses would be averaged for each range.

This is purely for fun, so there is one "rule." Researching is forbidden until all results are in. Even if you know how the results should be before they are in, they are to be kept a secret. Think of it as not spoiling the ending of a movie for someone who hasn't seen it.

So, if you'd like to take place in this experiment, just pick one of the ranges and begin. Use http://www.random.org/ (http://www.random.org/). Post your results here.

Thanks, and have fun!

[Ryli]

I'll only take part if the RNG is up to some betting lol

I'll start tomorrow.

[OldTrees]

1-10 is short enough of a range to calculate rather than test.
The theoretical average number of guesses for range 1-10 with the chosen number 1 is 5.5 exactly.
By symmetry the same is true of chosen number 10.

[Camoninja]

Looks like you are doing a binary search.
Math says that the RNG would at best guess it in 1 guess and at worst guess it in log base 2 of numbers to guess through turns.
There is a pattern to how likely each will be guessed, it's twice as likely to guess in two than one turn, three than two turns, etc. So you take the average of all the possibilities you just calculated and get a formula. I don't remember the formula right because I don't remember that day in my computer science class.

[OldTrees]

It is different from a binary search.
A binary search of 63 would be: 50,75,62,69,65,63

[QuantumT]

Is it required that we do it by hand? It seems like this is the kind of thing that a few lines of code would be well suited for.

[Camoninja]

It is different from a binary search.
A binary search of 63 would be: 50,75,62,69,65,63

Oh! You're right, it is.
Hm. I thought that the target number was always the same so I thought it would still be the same comcept. Now I realize it's both random so it isn't the same concept at all and it wouldn't be the same concept the way  saw it anyway.
Still, there should be a way to find the mathematical formula.

[agentflare]

Is it required that we do it by hand? It seems like this is the kind of thing that a few lines of code would be well suited for.

Agreed. The code needed would be pretty simple.

[Demagog]

Just as long as you use random.org, I don't care how you do it. Everyone has to use the same thing or the results could be wrong.